Advertisements
Advertisements
Question
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
Options
1
-1
2
-2
Advertisements
Solution
\[a - ib = \frac{3 - 4ix}{3 + 4ix}\]
\[ = \frac{3 - 4ix}{3 + 4ix} \times \frac{3 - 4ix}{3 - 4ix}\]
\[ = \frac{9 + 16 x^2 i^2 - 24xi}{9 - 16 x^2 i^2}\]
\[ = \frac{\left( 9 - 16 x^2 \right) - i\left( 24x \right)}{9 + 16 x^2}\]
\[ \Rightarrow \left| a - ib \right|^2 = \left| \frac{\left( 9 - 16 x^2 \right) - i\left( 24x \right)}{9 + 16 x^2} \right|^2 \]
\[ \Rightarrow a^2 + b^2 = \frac{\left( 9 - 16 x^2 \right)^2 + \left( 24x \right)^2}{\left( 9 + 16 x^2 \right)^2}\]
\[ = \frac{81 + 256 x^4 - 288 x^2 + 576 x^2}{\left( 9 + 16 x^2 \right)^2}\]
\[ = \frac{81 + 256 x^4 + 288 x^2}{\left( 9 + 16 x^2 \right)^2}\]
\[ = \frac{\left( 9 + 16 x^2 \right)^2}{\left( 9 + 16 x^2 \right)^2}\]
\[ = 1\]
Hence, the correct option is (a).
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib: (1 – i)4
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
Express the following complex number in the standard form a + i b:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[(1 + 2i )^{- 3}\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
If z1, z2, z3 are complex numbers such that \[\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1\] then find the value of \[\left| z_1 + z_2 + z_3 \right|\] .
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
Find a and b if a + 2b + 2ai = 4 + 6i
Express the following in the form of a + ib, a, b ∈ R i = `sqrt(−1)`. State the values of a and b:
`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
Evaluate the following : i888
Evaluate the following : i93
Evaluate the following : i403
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
State True or False for the following:
The order relation is defined on the set of complex numbers.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
